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Optimal transport and dynamics of expanding circle maps acting on measures

Published online by Cambridge University Press:  07 February 2012

BENOÎT KLOECKNER
BENOÎT KLOECKNER*
Affiliation:
UJF-Grenoble 1, CNRS UMR 5582 Institut Fourier, Grenoble, F-38401, France
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Abstract

In this paper we compute the derivative of the action on probability measures of an expanding circle map at its absolutely continuous invariant measure. The derivative is defined using optimal transport: we use the rigorous framework set up by Gigli to endow the space of measures with a kind of differential structure. It turns out that 1 is an eigenvalue of infinite multiplicity of this derivative, and we deduce that the absolutely continuous invariant measure can be deformed in many ways into atomless, nearly invariant measures. We also show that the action of standard self-covering maps on measures has positive metric mean dimension.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 2 , April 2013 , pp. 529 - 548
Copyright
©2012 Cambridge University Press

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References

[AGS08]Ambrosio, L., Gigli, N. and Savaré, G.. Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn(Lectures in Mathematics ETH Zürich). Birkhäuser, Basel, 2008.Google Scholar
[Bal00]Baladi, V.. Positive Transfer Operators and Decay of Correlations (Advanced Series in Nonlinear Dynamics, 16). World Scientific, River Edge, NJ, 2000.Google Scholar
[Gig09a]Gigli, N.. On the inverse implication of Brenier–McCann theorems and the structure of (P 2(M),W 2), available at http://cvgmt.sns.it/people/gigli/, 2009.Google Scholar
[Gig09b]Gigli, N.. Second order analysis on (P 2(M),W 2), Mem. Amer. Math. Soc., to appear. Available at http://cvgmt.sns.it/people/gigli/, 2009.Google Scholar
[KH95]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54). Cambridge University Press, Cambridge, 1995, with a supplementary chapter by Katok and Leonardo Mendoza.Google Scholar
[Klo10]Kloeckner, B.. A generalization of Hausdorff dimension applied to Hilbert cubes and Wasserstein spaces. Preprint, 2011, arXiv:1105.0360.Google Scholar
[Krz77]Krzyżewski, K.. Some results on expanding mappings. Dynamical Systems, Vol. II—Warsaw (Astérisque, 50). Soc. Math. France, Paris, 1977, pp. 205–218.Google Scholar
[Lot08]Lott, J.. Some geometric calculations on Wasserstein space. Comm. Math. Phys. 277(2) (2008), 423–437.Google Scholar
[LW00]Lindenstrauss, E. and Weiss, B.. Mean topological dimension. Israel J. Math. 115 (2000), 1–24.Google Scholar
[Ott01]Otto, F.. The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Differential Equations 26(1–2) (2001), 101–174.Google Scholar
[PSS03]Peres, Y., Simon, K. and Solomyak, B.. Fractals with positive length and zero Buffon needle probability. Amer. Math. Monthly 110(4) (2003), 314–325.Google Scholar
[Vil03]Villani, C.. Topics in Optimal Transportation (Graduate Studies in Mathematics, 58). American Mathematical Society, Providence, RI, 2003.Google Scholar
[Vil09]Villani, C.. Optimal Transport: Old and New (Grundlehren der Mathematischen Wissenschaften, 338). Springer, Berlin, 2009.Google Scholar